1. Field of the Invention
This invention relates to an improvement in a high gain feedback control system.
2. Description of the Prior Art
It has been widely known in the art that in a feedback control system the increase in gain of the system generally exhibits advantages such as the increase in response speed, the decrease in steady-state error and the like. Such advantages will be detailedly described hereinafter with reference to FIGS. 1 and 2 which illustrate a conventional high gain feedback control system.
In FIG. 1, reference characters X(s), D and E(s) respectively indicate a reference input, a summing point and a deviation signal. Reference numerals 1, 2 and 3 designate a high gain controller, a controlled system and a detecting means, respectively. Y(s) designates a controlled variable. Also, reference characters G.sub.1 (s), G.sub.2 (s) and G.sub.3 (s) designate transfer functions of the controller 1, controlled system 2 and detecting means 3, respectively. These functions are assumed to have characteristics represented by the following equations: ##EQU1## wherein K.sub.1, K.sub.2 and K.sub.3 indicate gains of the controller 1, controlled system 2 and detecting means 3, respectively; and T.sub.2 and s designate a time constant and an operator (d/dt), respectively.
In the conventional high gain feedback control system constructed in the manner as described above, a closed loop transfer function G.sub.01 (s) of from the reference input X(s) to the controlled variable Y(s) is represented by the following equation (2): ##EQU2## The substitution of the equation (1) for the equation (2) allows the following equation (3) to be obtained: ##EQU3##
In the equation (3), when a transfer function of the control system which is capable of rendering the gain K.sub.1 high to meet a relationship of K.sub.1 .multidot.K.sub.2 .multidot.K.sub.3 &gt;&gt;1 is designated by G.sub.02 (s), it is represented by the following equation (4) irrespective of characteristics of the transfer function G.sub.2 (s): EQU G.sub.02 (s)=1/K.sub.3 ( 4)
Thus, the controlled variable Y(s) is obtained by an equation Y(s)=X(s)/K.sub.3. This shows that the conventional high gain feedback control system can fully follow in a manner to allow its response speed to be significantly increased because there is no time lag in the reference input X(s).
Also, the deviation signal E(s) and the transfer characteristics .epsilon.(s) from the reference input X(s) to the deviation signal E(s) are represented by the following equations (5) and (6), respectively: ##EQU4## This shows that when the gain K.sub.1 is rendered high sufficiently to permit the closed loop transfer function G.sub.01 (s) to be equal to 1/K.sub.3, as the transfer characteristics G.sub.02 (s) in the equation (4); the transfer characteristics .epsilon.(s) is caused to be zero in the equation (6). This allows the control system to have a steady-state error of zero.
As can be seen from the foregoing, in the conventional high gain feedback control system, when the controlled system 2 having a time lag (time constant: T.sub.2) is provided with a high gain feedback, the closed loop control system including the controlled system 2 is allowed to have its response speed significantly increased to render the steady-state error of the control system substantially small. Nevertheless, the conventional high gain feedback control system is encountered with a problem that it often exhibits instability in the actual operation which never allows the control system to carry out the normal operation. Such instability would be due to the fact that a time lag which control elements included in the control system possess is neglected.
The conventional high gain feedback control system will be further described with reference to FIG. 2 which is a block diagram obtained by modifying the block diagram of FIG. 1.
In a control system of FIG. 2, assuming that the transfer function G.sub.3 (s) and gain K.sub.3 of a detecting means are assumed to be equal to 1 (G.sub.3 (s)=K.sub.3 =1), the transfer function G.sub.10 (s) of a controller 1' is represented by the following equation (8): ##EQU5## wherein K.sub.10 and T.sub.10 respectively indicate gain and time constant of the controller 1'. The transfer function G.sub.03 (s) of a closed loop system in the control system of FIG. 2 having such construction as described above is represented by the following equation (9): ##EQU6## Further, the characteristic equation of the control system is represented by the following equation (10): ##EQU7## wherein K is indicated by an equation K=K.sub.10 .multidot.K.sub.2. The second term of the equation (10) indicates a loop transfer function which is adapted to be utilized for the consideration of stability or instability of the feedback control system. In general, a loop transfer function of a feedback control system is represented by a total product obtained by multiplying all transfer functions included in the closed loop of the control system.
In the present case, there is no root indicating zero point of root loci in the second term of the equation (10) which expresses the loop transfer function. Whereas, roots indicating the poles are -1/T.sub.10 and -1/T.sub.2, respectively. Based on these facts, a consideration will now be made on root loci of the above-mentioned loop transfer function obtained when the value of the gain K described above is varied from zero to infinity on an s-plane which is a complex plane in a root locus stability criterion known as one of stability criteria for a high gain feedback control system. First, assuming that the value of K is zero, the roots indicating poles are -1/T.sub.10 and -1/T.sub.2 ; thus, these two poles constitute the respective starting points of the root loci. As the value of K is gradually increased toward infinity, the loop transfer function forms root loci extending from the above-mentioned two poles along positive and negative imaginary axes to infinity, respectively. Thus, it will be noted that the conventional high gain feedback control system has a disadvantage that it is unstable sufficient to exhibit natural frequency because its damping ratio becomes zero at the time when the value of K is infinite.